Zeta-function regularization of infinite series is the most commonly used in QFT applications. However, occasionally other schemes are employed which, allegedly, suit the nature (most noticeably the underlying symmetries) of the problem at hand.
An alternative to the zeta-function (used for instance at the bottom of Page 6 in hep-th/0311021) is to sum all the terms "on equal footing", where the quoted phrase refers to the fact that some other methods employ regulators (e.g. $e^{-m\epsilon}$ for the $m$'th term) to exponentially (or otherwise) suppress the consequitive terms, and then throw away the infinities that turn up while removing the regulator by taking limits. This alternative, on the other hand, just sums the series up to some cutoff $N$ and then removes the $N$-dependent parts. For example, the sum of integers would be zero with this regularization, as $N(N+1)/2$ has no $N$-independent part.

I would appreciate it if someone can elaborate on the issue, by explaining why this alternative regularization scheme respects supersymmetry (as claimed in the provided reference) but the zeta-function regularization does not.

There is a very nice (but not substantiated) discussion on Zeta respecting Conformal and Modular invariance in relation with String theory applications here: motls.blogspot.com/2007/09/…
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ArashOct 15 '12 at 5:26

But unfortunately, the author gets lost in his desire for making moral and philosophical manifestos, instead of communicating his much needed isight into the regularization issue. For instance, it is not mentioned, how local counter terms play a role, or how, in the general case, one can identify the symmetry-consistent regularization schemes.
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ArashOct 15 '12 at 5:30